Group in which every subgroup is powering-invariant

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group in which every subgroup is powering-invariant if it satisfies the following equivalent conditions:

  1. It is a group in which every subgroup is a powering-invariant subgroup.
  2. It is either a periodic group or it is not powered over any prime.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite group |FULL LIST, MORE INFO
periodic group every element has finite order |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every normal subgroup is powering-invariant every normal subgroup is powering-invariant. |FULL LIST, MORE INFO
group in which every characteristic subgroup is powering-invariant every characteristic subgroup is powering-invariant. |FULL LIST, MORE INFO
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